LDAUTYPE

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LDAUTYPE = 1 | 2 | 4
Default: LDAUTYPE = 2 

Description: LDAUTYPE specifies which type of L(S)DA+U approach will be used.

  • LDAUTYPE=1: The rotationally invariant LSDA+U introduced by Liechtenstein et al.[1]
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): E_{{{\rm {HF}}}}={\frac {1}{2}}\sum _{{\{\gamma \}}}(U_{{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}}-U_{{\gamma _{1}\gamma _{3}\gamma _{4}\gamma _{2}}}){{\hat n}}_{{\gamma _{1}\gamma _{2}}}{{\hat n}}_{{\gamma _{3}\gamma _{4}}}
and is determined by the PAW on site occupancies
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {{\hat n}}_{{\gamma _{1}\gamma _{2}}}=\langle \Psi ^{{s_{2}}}\mid m_{2}\rangle \langle m_{1}\mid \Psi ^{{s_{1}}}\rangle
and the (unscreened) on site electron-electron interaction
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): U_{{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}}=\langle m_{1}m_{3}\mid {\frac {1}{|{\mathbf {r}}-{\mathbf {r}}^{\prime }|}}\mid m_{2}m_{4}\rangle \delta _{{s_{1}s_{2}}}\delta _{{s_{3}s_{4}}}
where |m⟩ are real spherical harmonics of angular momentum L=LDAUL.
The unscreened e-e interaction Uγ1γ3γ2γ4 can be written in terms of the Slater integrals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): F^{0} , , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): F^{4} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): F^{6} (f-electrons). Using values for the Slater integrals calculated from atomic orbitals, however, would lead to a large overestimation of the true e-e interaction, since in solids the Coulomb interaction is screened (especially Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): F^{0} ).
In practice these integrals are therefore often treated as parameters, i.e., adjusted to reach agreement with experiment in some sense: equilibrium volume, magnetic moment, band gap, structure. They are normally specified in terms of the effective on site Coulomb- and exchange parameters, U and J (LDAUU and LDAUJ, respectively). U and J are sometimes extracted from constrained-LSDA calculations.
These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): L\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): F^{0}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): F^{2}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): F^{4}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): F^{6}\;
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 1\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): U\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 5J\; - -
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 2\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): U\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\frac {14}{1+0.625}}J Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 0.625F^{2}\; -
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 3\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): U\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\frac {6435}{286+195\cdot 0.668+250\cdot 0.494}}J Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 0.668F^{2}\; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): 0.494F^{2}\;
The essence of the L(S)DA+U method consists of the assumption that one may now write the total energy as:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): E_{{{\mathrm {tot}}}}(n,{\hat n})=E_{{{\mathrm {DFT}}}}(n)+E_{{{\mathrm {HF}}}}({\hat n})-E_{{{\mathrm {dc}}}}({\hat n})
where the Hartree-Fock like interaction replaces the L(S)DA on site due to the fact that one subtracts a double counting energy (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): E_{{{\mathrm {dc}}}} ) which supposedly equals the on site L(S)DA contribution to the total energy,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): E_{{{\mathrm {dc}}}}({\hat n})={\frac {U}{2}}{{\hat n}}_{{{\mathrm {tot}}}}({{\hat n}}_{{{\mathrm {tot}}}}-1)-{\frac {J}{2}}\sum _{\sigma }{{\hat n}}_{{{\mathrm {tot}}}}^{\sigma }({{\hat n}}_{{{\mathrm {tot}}}}^{\sigma }-1).
  • LDAUTYPE=2: The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev et al.[2]
  • LDAUTYPE=4: same as LDAUTYPE=1, but LDA+U instead of LSDA+U (i.e. no LSDA exchange splitting).

Related Tags and Sections

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT

References

  1. ↑ A. I. Liechtenstein, V. I. Anisimov and J. Zaane, Phys. Rev. B 52, R5467 (1995).
  2. ↑ S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).

Contents

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